7x - 5y = 44x - 8y = 9If \((\mathrm{x}, \mathrm{y})\) is the solution to the system of equations above,...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{7x - 5y = 4}\) (first equation)
  • \(\mathrm{4x - 8y = 9}\) (second equation)
• Find: The value of \(\mathrm{3x + 3y}\)

2. INFER the most efficient approach

• Key insight: Instead of solving for x and y individually, we can manipulate the given equations to directly obtain an expression for \(\mathrm{3x + 3y}\)
• Strategy: Look for a way to combine the equations that will result in coefficients of 3 for both x and y

3. SIMPLIFY by subtracting the equations

• Subtract the second equation from the first equation:
  \(\mathrm{(7x - 5y) - (4x - 8y) = 4 - 9}\)

• Distribute the negative sign carefully:
  \(\mathrm{7x - 5y - 4x + 8y = -5}\)

• SIMPLIFY by combining like terms:
  \(\mathrm{(7x - 4x) + (-5y + 8y) = -5}\)
  \(\mathrm{3x + 3y = -5}\)

Answer: -5 (Choice B)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when distributing the negative sign during equation subtraction.

They might incorrectly write: \(\mathrm{(7x - 5y) - (4x - 8y) = 7x - 5y - 4x - 8y = -5}\), forgetting that subtracting \(\mathrm{-8y}\) gives \(\mathrm{+8y}\). This leads to \(\mathrm{3x - 13y = -5}\), which doesn't directly give them \(\mathrm{3x + 3y}\) and causes confusion about how to proceed. This leads to guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the direct manipulation strategy and instead try to solve for individual x and y values.

This approach involves more complex elimination or substitution steps that can lead to calculation errors and fractions. Even if done correctly, it's unnecessarily time-consuming. When students get bogged down in lengthy calculations, they may make arithmetic mistakes or run out of time, leading to incorrect answer selection.

The Bottom Line:

This problem rewards strategic thinking over computational power. The key insight is recognizing that equation manipulation can directly produce the desired expression without finding individual variable values.